Transition between SZ and SD states

We investigate the role of cell size and relative (to intracellular volume) extracellular space on neuronal behavior by varying the radius of the cell, r_in, keeping the total radius of cell and extracellular space, r_tot, fixed. Thus changing r_in is largely equivalent to changing the ratio of intra- to extracellular volume. Since we are interested in the pathological states of the cell, we use K⁺ concentration in the distant reservoir [K]_o,∞ = 8mM—a value typically used for inducing SZ in in vitro [30]. The behavior of the cell changes dramatically as we increase r_in. A bifurcation diagram showing the maximum and minimum of [K]_o as a function of r_in is shown in Fig 1A. Briefly, we fixed r_in and ran the simulation generating a time-trace representing [K]_o versus time. Depending on the r_in value, [K]_o either oscillates or converge to a steady state value. The initial few hundred seconds of the time-trace were discarded as a transient period and the maximum and minimum of [K]_o values in the remaining trace were recorded. The lower and upper markers (green) respectively at a given r_in in Fig 1A represent the maxima and minima of [K]_o oscillations for that particular value of r_in. This process was repeated many times, each time incrementing r_in by a small amount. For r_in values, where [K]_o does not oscillate (i.e. [K]_o converges to a steady state), the maxima and minima have the same value and is represented by a line (stable, red; unstable, blue). To capture both stable and unstable behaviors, the steady states were simulated in XPPAUT.

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Fig 1. Cell shows a variety of behaviors as we vary its size.

Here we consider Vol (Eq 14) as a bifurcation parameter and simulate Eqs (1, 3, 5, 7, 10). (A) Bifurcation diagram of [K]_o as a function of r_in where green circles, red solid line, and blue dashed line represent periodic orbit, stable, and unstable steady states respectively. The four regions in the bifurcation diagram marked by SZ, SD, and RS represent the parameter-regions where seizure, spreading depression, and resting states are observed respectively. The black solid and dashed limit cycles represent the change in r_in as different ion concentrations vary during a single SZ at [K]_o,∞ = 8mM and 9mM respectively. For the limit cycles, the instantaneous r_in values in the limit cycles are obtained from Eq (13). The four panels on the right show membrane potential (black), reversal potential of K⁺ (blue), Cl⁻ (green), and Na⁺ (red) of the cell with different r_in values given in the right corner of each panel.

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For all r_in < 4.66μm, [K]_o remains unchanged and the cell remains in steady state with V ≈ −60mV (Fig 1B). As we increase r_in above 4.66μm, [K]_o enters a periodic orbit via a Hopf bifurcation and starts oscillating with a small amplitude and the cell exhibits spontaneous periodic SZs (Fig 1C) similar to those observed in experiments [36]. There is a sudden increase in the amplitude of [K]_o oscillations at r_in = 4.826μm where the peak [K]_o goes well over 26mM—a concentration that is often used for inducing SD [37]. The periodic SZs transform to a behavior where the cell is locked into a depolarized state after burst-spiking and exhibits a few small-amplitude spikes on its way out of the depolarized state (Fig 1D). As we increase r_in further, this state disappears making way for mixed SZ and SD behavior where the high-frequency spiking is followed by the locking of V into a depolarized state and the cell comes out of the depolarized state without spiking (Fig 1E). Such mixed states are typically seen in the cells in hypoxic SD [32] or immature physiological conditions [38]. It is worth mentioning that this locking of neuronal membrane into depolarized state is the condition for SD at the single cell level [39]. At the network or tissue level the depolarization may also propagate [40]. This unification of SZ and SD dynamics is supported by the increasing discovery of monogenic mutations in humans that lead to both SZs and migraines [41]. The cell exhibits SZ-SD mixed behavior until it makes a transition to a silent state via another Hopf bifurcation at r_in > 4.924μm where V remains fixed at a stable resting value.

The bifurcation diagrams for [Na]_i, [K]_i, and [Cl]_i (similar to Fig 1A) (not shown) show that their behavior contrasts with [K]_o. That is, the amplitude (defined as the difference between the maximum and minimum) of [Na]_i, [K]_i, and [Cl]_i oscillations decreases as we increase r_in. At r_in > 4.924μm, the relatively larger intracellular volume dominates the extracellular space and these 3 concentrations drop to resting values leading to a return to resting membrane potential. That is, the extremely large intracellular volume leads to low intracellular concentrations that overshadow the effect of the changes in extracellular ion concentrations. The bifurcations in the cell’s behavior described above are qualitatively preserved when [K]_i, [Na]_o, and [Cl]_o are modeled by rate equations [33] instead of conservation equations (S1 Fig). It is important to point out that the kind of changes in r_in shown in Fig 1A and the rest of the paper are physiologically relevant. For example, the surface area of cortical layer V pyramidal neurons increases by more than 50% in response to OGD [20]. Thus a spherical cell with initial radius of 4.75μm would swell to a final radius of 5.81μm, shrinking the extracellular space significantly.

The change in the ratio of intra- to extracellular volume as a result of changing r_in plays a major role in the transition between SZ and SD states. Depending on the value of β, the cell exhibits steady state, SZ, or SD without any transition between these behaviors as we change r_in if β is kept constant (S2(A) Fig). Using β as a bifurcation parameter at fixed r_in on the other hand causes the cell to make the transitions between steady state, SZ, and SD (S2(B) Fig). Nevertheless, the cell size per se is an important parameter that together with β shapes the bifurcations and the parameter ranges where different behaviors are observed (compare Fig 1A and S2 Fig).

The results in Fig 1 indicate that the size of the cells and how tightly they are packed in the tissue can play a significant role in their dynamics. An important followup question would be: could a cell swell enough so that it would spontaneously go through the transitions shown in Fig 1? To answer this question, we add the volume dynamics given by Eq (14) to our model, where the cell volume depends on the instantaneous ion concentrations. We compute the spontaneous change in r_in as the ion concentrations inside and outside the cell vary during a single SZ (solid black line in Fig 1A). The limit cycle shows that r_in can change enough during one SZ so that the cell would make the transition from SZ to SD regions. A cell with r_in = 4.82μm (which is in the SZ region and will exhibit spontaneous SZs similar to Fig 1C) before a SZ starts would swell to a final r_in = 4.84μm (solid black line in Fig 1A), well within the SD region (Fig 1D). The crossover to the SD region is more prominent for higher K⁺ in the bath solution (dashed black line in Fig 1A).

Fig 2 shows the pathway to the spontaneous transition from SZ to SD caused by cell swelling obtained by simulating the full model (Eqs 1–15). The arrows in Figs 2, 6, and 9 indicate the direction of the trajectory. Initially [Na]_i and [K]_o slowly build up leading to an increase in intracellular volume (hence a decrease in the relative extracellular space) and higher excitability of the cell. The microenvironment reaches a point where V makes a transition from steady state to limit cycle via a saddle node on invariant cycle (SNIC) bifurcation and the cell exhibits a SZ (Fig 2A). After exiting the SZ state, the cell does not have sufficient time to reverse the swelling (normally pumps would restore ionic gradients reversing the swelling) and the cell exhibits a SD event by entering a second periodic orbit via a SNIC, with progressively decreasing amplitude followed by entrance into a depolarized state via a Hopf bifurcation. Fig 2B shows the variations in [Na]_i during this transition. The thick arrow in Fig 2A indicates that the cell would go into a depolarized state similar to ischemia-induced AD if [K]_o increases further either due to swelling or lack of O₂. This point will be further elaborated later in this paper.

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Fig 2. Transition of the cell from SZ to SD due to cell swelling.

(A) Changes in membrane potential, [K]_o, and r_in of the cell as it transitions from SZ to SD. (B) is from the same simulations as in (A) with the horizontal axis representing [Na]_i instead of V. The thick arrow indicates that for higher [K]_o, whether due to higher K⁺ exposure or hypoxia, the cell exhibits physiological AD as demonstrated below. Simulations based on the full model.

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What physiological mechanisms help to regulate the brain so that most of the time, even in people with chronic recurring seizures and migraines, their brains are operating normally? To begin to address this question we show a two-parameters bifurcation diagram for [K]_o in Fig 3 where the two parameters are r_in and glial K⁺ buffering strength, B_glia. In Fig 3A, we show the maximum of [K]_o as a function of r_in and B_glia. As is clear, for weaker glial buffering the cell’s behavior changes as a function of r_in in the same manner as in Fig 1. However, for stronger glial buffering (B_glia > 21mM/sec) the cell becomes biased towards SZ behavior and is less likely to go to SD. The stronger glial buffering siphons away [K]_o fast enough so that the cell recovers from swelling during SZ before entering SZ again. For even stronger glial buffering, the cell neither shows SZ nor SD behavior. This point is emphasized further in Fig 3B where we show the three regions described in Fig 1 as a function of r_in and B_glia. For larger B_glia, both SZ and SD regions disappear. This demonstrates that glial K⁺ buffering can play a neuroprotective role against SZ and SD behaviors and, if strong enough, will constrain the neuron to physiological dynamics. From our previous work, we know that extracellular diffusion shares similar dynamical properties with the glial buffer (see, e.g., Fig 7 in [18]).

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Fig 3. Two parameters bifurcation diagram for [K]_o.

(A) The maximum of [K]_o oscillations as a function of r_in and B_glia. (B) The regions where the cell shows resting state (RS), SZ, and SD behaviors. We simulate Eqs (1, 3, 5, 7, 10) and take volume as a bifurcation parameter.

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A two parameter bifurcation diagram for [O₂] from simulations in Fig 3 shows that the resting state at large r_in (right side in Fig 3B) is energetically favorable as compared to the one at small r_in (left side in Fig 3B) (Fig 4). Interestingly, the local O₂ consumption during SD in a cell with smaller radius is much larger than the consumption in a cell that has swollen. This indicates that although the [K]_o changes during SD in smaller cells are smaller, Na⁺−K⁺ exchange pumps consume more energy to restore physiological [K]_o and [Na]_i in these cells due to their relatively larger extracellular reservoirs. A cell with an infinitesimally small extracellular space requires an infinitesimally small ionic flux to substantially change its extracellular potential, and an infinitesimally small work load on the membrane pumps to restore those gradients. As mentioned above, the bifurcation diagrams for [Na]_i, [K]_i, and [Na]_o (similar to Fig 1A) (not shown) show that their behavior contrasts with [K]_o. That is, the amplitudes of [Na]_i, [K]_i, and [Na]_o oscillations decrease with increasing r_in. Because the pumps are stimulated by [Na]_i and [K]_o (Eq 3), higher concentrations of [Na]_i in a smaller cell overwhelms the relatively smaller concentrations of [K]_o and the Na⁺ − K⁺ exchange pumps increase their rates accordingly to rearrange those ions (Fig 4). It is important to point out that the amount of oxygen available to the cell also acts as a bifurcation parameter causing the cell to transition between different states [18, 33].

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Fig 4. Two dimensional bifurcation diagram for local available O₂ from simulations in Fig 3.

The vertical axis shows the minimum of [O₂] during oscillations.

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Bifurcation analysis of different neuronal behaviors

To gain further insights into the mechanisms behind the three behaviors in Fig 1C-1E exhibited by the cell, we performed a bifurcation analysis of the model fixing the slower variables [K]_o, [Na]_i, [Cl]_i, [O₂], and volume (Vol). In Fig 5, we show the maxima and minima of V as a function of [K]_o. For small [K]_o values, the cell remains in stable steady state (SSS1; red line on the left) which collides with an unstable steady state (USS; black dotted line) giving rise to a periodic orbit (PO; green circles) through a SNIC bifurcation (Fig 5A). The unstable steady state gains stability (SSS2; red line on the right) through a Hopf bifurcation (HB) where the membrane potential of the cell is locked into a depolarized state. The SNIC moves to the right as we increase [Na]_i (Fig 5B) and bifurcates into a saddle homoclinic (HC) bifurcation (Fig 5C,5D).

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Fig 5. Bifurcation diagrams showing V as a function of [K]_o for four different [Na]_i values.

(A) [Na]_i = 18mM, (B) [Na]_i = 24mM, (C) [Na]_i = 29mM, and (D) [Na]_i = 32mM. For the simulation in this figure, [Cl]_i = 8mM and normal [O]₂ of 30 mg/L is used. SSS, USS, LP, PO, and HC stand for stable steady state, unstable steady state, limit point, periodic orbit, and homoclinic bifurcation respectively.

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The locations of special points, HB, SNIC, the limit point (LP) (Fig 5A), and HC (Fig 5C) as [K]_o and [Na]_i vary simultaneously are shown through a two-parameter bifurcation diagram in Fig 6. The limit cycles and steady states from the full (but with fixed volume) model at different r_in values are also shown in Fig 6. For clarity, we will restrict our discussion to the case of [Cl]_i = 8mM but the argument applies to the other values of [Cl]_i as well. During the SZ event like the one shown in Fig 1C (r_in = 4.82μm), the cell enters the periodic orbit from steady state on the left (SSS1 in Fig 5A) through a SNIC bifurcation and spikes for a few seconds before going back to SSS1. The [K]_o values when the trajectory crosses the SNIC bifurcation curve indicate the beginning and end of the burst in the time trace. Since the burst terminates at a SNIC, the frequency of the burst is low at the beginning and end of the burst. This limit cycle with bursts never generates [K]_o substantially above a physiological ceiling, about 12 mM experimentally [42] and about 13–14 mM in Fig 6, below which spikes and seizures are observed but not SD. Indeed, the model generates rather pure seizure dynamics with [K]_o below the ceiling.

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Fig 6. Two-parameter bifurcation diagram.

The locations of HB (red), SNIC (blue), HC (pink), and LP (green) with [K]_o and [Na]_i as bifurcation parameters for three different [Cl]_i values and normal [O]₂ of 30 mg/L. The locations of SNIC and HC change with intracellular Cl⁻ as shown for [Cl]_i = 6mM, 8mM, and 10mM where the thickness of the blue and pink lines represents increasing [Cl]_i value. The position of HB and LP does not change significantly and is therefore only shown for [Cl]_i = 8mM. The three solid black traces represent the limit cycles from the full model (Eqs 1, 3, 5, 7, 10), and fixed volume) for three r_in values shown in the figure in μm and correspond to the three behaviors in Fig 1C (with slightly larger r_in), Fig 1D and 1E respectively. The red arrows show the direction of trajectories. The diamond and small circle represent steady states at r_in = 4.6μm and 4.95μm respectively. The black dashed curve is a 2D version of Fig 2B and is shown for comparison.

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At r_in = 4.84μm (Fig 1D), there is a substantial change in [Na]_i and [K]_o and the one-parameter bifurcation diagram goes through the transitions shown in Fig 5A-5D. The beginning of the first burst starts when trajectory crosses the SNIC curve in Fig 6 and terminates when it crosses the HB curve. Again, the frequency within the burst is low at the beginning. The amplitude shrinks to zero at the end as the trajectory passes through the HB curve. After the first burst terminates, the trajectory follows the stable branch of SSS2 (Fig 5) for a while before it crosses the HB curve again. The solution does not start to burst at this moment, because of the delayed loss of stability phenomenon (delayed HB) [43] that occurs when a parameter passes slowly through a HB point and the system’s response changes from a slowly varying steady state to a slowly varying oscillation. So, the trajectory traces the unstable branch of SSS2 for a while before it starts to burst between HB and HC curves. Also referred to as “delayed” or “memory effect”, this transition happens when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects the dynamic aspect of the parameter variation. This memory effect has been studied for different problems including the FitzHugh-Nagumo model [44]. The second burst terminates when the trajectory crosses the HC curve (notice the slight delay in the termination of the second burst for higher [Cl]_i values). At this moment, the solution drops to the lower stable branch SSS1. Since [Na]_i is approximately 30mM for the second burst, the amplitude of the second burst is smaller according to Fig 5. The delay also explains why the burst onset has a nonzero amplitude since the solution is away from the HB curve. The amount of delay depends on the time spent near the attracting branch of SSS2 (the loop from HB to HB).

The explanation for r_in = 4.84μm also applies to the case with r_in = 4.9μm (Fig 1E), except that the HB to HB trajectory loop is more dramatic than r_in = 4.84μm. The trajectory spends more time on the attracting branch of SSS2 that increases the delay. Here, the solution moves along the unstable branch through the entire HB-HC regime without oscillating. It finally loses stability, but since there is no stable limit cycle anymore, it can only drop to the lower stable branch of SSS1. SSS2 (the depolarized state of V) ends to the left of HC curve. The drop to SSS1 (end of depolarized state) is faster (as in Fig 1D) for a slightly smaller cell (for example when r_in = 4.88μm) because the trajectory is close to the SNIC-HC bifurcation point and hence the upper and lower branches are closer (not shown).

Anoxic depolarization

In addition to SZ, SD, and SZ-SD transition, our model closely reproduces AD. AD, the initial electrophysiological event during ischemia, is a front of depolarization that drains residual stored energy in compromised gray matter. Recently, Brisson and Andrew studied AD at the single cell level in neocortex and hypothalamus slices deprived of O₂ and glucose [20]. They used two-photon microscopy to image cell swelling simultaneously with patch-clamp membrane potential measurements in the OGD condition. Although, our model does not have the glucose component, it behaves in the same manner as that observed experimentally during energy substrate deprivation. We induced energy deprivation (ED) in the model by putting oxygen in the perfusion solution, [O₂]_∞, equal to zero, which is model-equivalent to OGD or ischemia-induced AD [45]. In Fig 7A we show the cell behavior in response to 5 min ED. During AD, the cell swells qualitatively in the same manner as observed in experiments (Fig 7B and 7E). The extra- and intracellular ion concentrations go through a massive redistribution during AD (Fig 7C). Consistent with the experimental observations (shown in panels D-F), the cell depolarizes and swells significantly during AD. The behavior of the model-cell is reminiscent of the magnocellular neuroendocrine cell (MNC) in the hypothalamus with weak AD that depolarizes only transiently close to 0mV (Fig 7D). However, as we change the initial radius of the cell from 4.0μm to 3.0μm, it exhibits strong AD, approaching a steady Donnan equilibrium of −2mV (red line in Fig 7A). This behavior is reminiscent of strong AD in pyramidal neurons from neucortical layer V where the cell’s membrane depolarizes to near 0mV during OGD and does not recover to the pre-OGD state when O₂ and glucose supply is restored (Fig 7F). Our result sheds light on the importance of cell size and packing density in the strength of AD and the potential damage to the cell where the membrane potential is permanently brought into a depolarized state with no recovery to the pre-ED state.

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Fig 7. Model cell response to energy deprivation (ED, the model-equivalent of OGD).

The cell exhibits AD in response to 5 min ED and returns back to normal behavior after ED ends (A). Change in cell radius (B) and ion concentrations (C) during ED. Solid, dashed, and dotted lines in (C) represent [K]_o, [Na]_i, and [Cl]_i respectively. The black line in (A) and all lines in (B, C) correspond to initial r_in = 4μm, while the red line in (A) is for initial r_in = 3μm. (D) shows experimental membrane potential of a magnocellular neuroendocrine cell (MNC) under 15 min OGD. Panel (E) shows the percent change in light transmittance (ΔLT) representing cell swelling, while (F) shows AD exhibited by a pyramidal neuron in neucortical layer V. (D-F) Data provided by David Andrew. (D-F) modified from [20] with permission American Physiological Society. Simulations based on the full model.

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During the modeling of AD, we made some additional observations. The main conclusion about the transition between SZ and SD states qualitatively remain the same for fixed Cl⁻ concentrations. However, the cell cannot sustain a prolonged depolarization similar to AD for fixed Cl⁻ concentrations (S3(A) Fig), confirming an important role of Cl⁻ in AD [46]. A similar behavior is observed when we maintain the normal K⁺ diffusion between extracellular space and bath solution (S3(B) Fig). Because diffusion of ions within the extracellular space is a function of the size and geometry of extracellular space [47], we needed to make diffusion a function of O₂ [33] so that there is decreased exchange of K⁺ between bath or blood vessels and extracellular space when the cell is exposed to reduced oxygen. In addition, we had to make the K⁺ glial buffering a function of O₂ to reproduce AD (as in [48]). Several observations support these assumptions. A significant portion of [K]_o is buffered by astrocytes through ATP-dependent Na⁺ − K⁺ pumps that do not function in the absence O₂ and glucose. In the absence of O₂, astrocytes attempt to buffer the increased extracellular K⁺ by switching to anaerobic glycolysis and swell substantially [49], further restricting K⁺ diffusion and limiting glial energy reserves. Astrocytic inward rectifying K⁺ channels (K_ir) also contribute to K⁺ siphoning, gating through interaction with G-protein coupled receptors pathway that is dependent on ATP (see [50] for a review on K_ir channels). Similarly, Na⁺ − K⁺ − Cl⁻ cotransporters (NKCC) that are found in astrocytes play a significant role in transferring K⁺ (together with Na⁺ and Cl⁻) from extracellular space to astrocytes and are dependent on ion gradients [51] and thus indirectly on ATP. Hence ATP plays a crucial role in these pathways that would be disrupted in the absence of O₂, leading to reduced K⁺ buffering.

The other important effect is the reduced transport of ions at glial end-feet adjoining the pericapillary space. The two-way K⁺ trafficking at the blood-brain barrier occurs at the junctions between astrocytic end-feet and blood vessels (see for example [52]). Astrocytes release K⁺ next to tight junction sealed endothelial cells in blood vessels. Na⁺ − K⁺ pumps transfer that K⁺ to the endothelial cells and it is then delivered into the blood vessels through K⁺ channels. A reverse process transfers K⁺ from blood vessels to astrocytes and finally to neurons (Fig 8). The lack of ATP in AD would disrupt this pathway, consequently impairing the K⁺ diffusion between blood vessels and extracellular space.

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Fig 8. K⁺ exchange at the blood-brain barrier.

Extracellular K⁺ absorbed by astrocytes is transferred to the blood vessels through a sequential functioning of K⁺ channels and Na⁺ − K⁺ exchange pumps at the junction between astrocytes and endothelial cells surrounding the blood vessel lumen. A reverse process transfers K⁺ from blood vessels to astrocytes. Omitted from this picture is the K⁺ exchange between astrocytes through gap junctions.

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In Fig 9, we show the pathways of microenvironment changes leading to AD. Immediately after setting O₂ to zero, there is a rapid increase in [K]_o and [Na]_i followed by a slow increase in intracellular volume (Fig 9A). [K]_o drops slightly from its peak value once it enters the AD due to the slight delay in the [Cl]_i rise (see Fig 7C). After the initial drop, [K]_o stabilizes at fixed value when [Cl]_i plateaus at its peak value. Once O₂ is restored, [K]_o is restored to normal values followed by slow restoration of [Na]_i and intracellular volume. The blue curve is replotted from Fig (2B) to compare the cell dynamics during SZ, SD, and AD. The change in the membrane potential along with [K]_o and percentage change in the cell volume during the simulation represented by the dashed line in Fig 9A is shown in Fig 9B. The gray planes in Fig 9A indicate the approximate regions for SZ, SD, and AD. The cell exhibits SZ below the physiological [K]_o ceiling represented by the bottom plane, SD between bottom and top planes, and AD above top plane. The position of these planes will change with the size and density of neurons.

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Fig 9. A comparison of microenvironmental changes during SZ, SD, and AD.

(A) Solid and dashed lines represent the simulations in Figs (2) and (7) respectively. %ΔVol represents the percent change in volume and is defined as , where r_in,ss and r_in,ins represent initial steady state (ss) and instantaneous (ins) radius of the cell respectively. The peak [K]_o values during SZ, SD, and AD are shown for comparison. The lower grey plane corresponds to the physiological ceiling for [K]_o [42], here calculated at 12.9 mM in the model, whereas the upper plane, which separates SD from AD, is found at 39 mM [K]_o. (B) is from the same simulations as shown by the dashed line in (A) and shows the change in membrane potential along with [K]_o and volume. Simulations based on the full model.

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Hodgkin-Huxley type equations

The membrane potential V of the cell is modeled with the following set of modified Hodgkin-Huxley type equations [15, 63] (1) Where n⁴ and m³ h represent the gating variables for delayed rectifier potassium (I_K) and transient sodium (I_Na) currents. The leak current (I_L) has three components: K⁺ (I_KL), Na⁺ (I_NaL), and chloride (I_ClL) leak. I_pump is the net current due to the ATP-dependent pump that extrudes 3 Na⁺ for bringing 2 K⁺ in. A random current (I_rand) representing the background input from the other neurons is included in some simulations and is modeled by a zero mean Gaussian processes. The meaning and values of parameters used in the model are given in Table 1.

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Table 1. Model Parameters.

https://doi.org/10.1371/journal.pcbi.1004414.t001

The rate equations for the gating variables are [64] (2) We will use the instantaneous steady-state form of m, i.e. m = α_m/(α_m + β_m) [65].

Ion concentration dynamics

The ion current equations are augmented with dynamic intra- and extracellular concentrations of K⁺, Na⁺, and Cl⁻. These concentrations are modulated by the neuron’s intrinsic ionic currents, Na⁺ − K⁺ pump current, and K⁺ − Cl⁻ cotransporters. The glial buffering and diffusion into the microenvironment of the cell (from the bath solution in slice preparation and vasculature in vivo) also modulate the K⁺ concentration in the extracellular space. The ion concentrations inside and outside the cell are coupled to the membrane voltage equations via the Nernst equation. The rate equations for K⁺ and Na⁺, O₂, and Cl⁻ concentrations and the rational for different terms within these equations are described in detail in [15, 66–68], [18], and [33] respectively and are summarized below.

Given I_K, I_KL, I_pump, diffusion of K⁺ to the microenvironment (I_diff), glial buffering (I_glia), and K⁺ − Cl⁻ cotransporters (I_KCC), the extracellular K⁺ dynamics, [K]_o, can be represented in the model as (3) where τ = 1000 is used to convert seconds to milliseconds. γ converts ionic current to concentration rate of change and is calculated using [15], where A, Vol, and F represent cell area, volume, and Faraday constant respectively. β is the intra- to extracellular volume ratio. γ and β change when the volume and surface area of the cell change. In simulations where volume is treated as a dynamical variable, both γ and β change dynamically. Modeling glial K⁺ buffering by a rate equation as done in [16] qualitatively did not change our results. Similarly, making I_diff a function of β explicitly as done previously [33] did not change our conclusions (but see the discussion in “Anoxic depolarization” section).

The Na⁺ − K⁺ pump is modeled as a product of sigmoidal functions, ρ is the maximum pump strength, and [Na]_i is the intracellular Na⁺ concentration [15]. Each sigmoidal term saturates for high values of [Na]_i and [K]_o respectively. The ATP required to keep the pump running depends on the local O₂ availability. The ATP concentration and hence the pump strength decreases as the cell depletes its local O₂ reservoir. We use a sigmoid function 𝓕([O₂]) to model the O₂ concentration ([O₂])-dependence of the pump activity [18] (4) The justification for the [O₂]-dependence of I_diff and I_glia terms is elaborated in [33] and further discussed in the Results section. For simplicity, we use the same functional form as in I_pump for the [O₂]-dependence of I_diff and I_glia. Using separate functions for the [O₂]-dependence of these three fluxes as in [33] does not qualitatively change our results.

The local O₂ concentration is controlled by the activity of the Na⁺ − K⁺ pump and diffusion of O₂ from the bath solution (or vasculature) to the extracellular space. Thus the rate equation for [O₂] as developed in [18] based on [O₂] imaging experiments [17] is given as (5) Where [O₂]_∞ is the oxygen concentration in the perfusion solution with a normal range of 30 − 32mg/L when aerated with 95%O₂ and 5%CO₂ at 32–34°C. α is a conversion factor that converts pump current in mM/s to O₂ concentration change in mg/L/s and the diffusion constant of O₂ (ϵ_O) is obtained from Fick’s law [18].

[K]_o,∞ in the diffusion equation (Eq 3) is the K⁺ concentration in the nearby reservoir. Physiologically, this would correspond to either the bath solution in a slice preparation, or the vasculature in the intact brain (noting that [K]_o is kept below the plasma level by trans-endothelial transport). In physiological conditions, [K]_o,∞ is set equal to 4mM. We use a previously published value for the diffusion constant of [K]_o to the nearby reservoir ϵ_K [66] which was obtained from Fick’s law. That is, ϵ_K = 2D/Δx², where D = 250 × 10⁻⁶cm²/sec is the K⁺ diffusion constant in neocortex [69], and Δx ≈ 20μm for brain reflects the average distance between neurons and capillaries [70].

Both active and passive K⁺ uptake into glia is incorporated into a simplified single sigmoidal response function that depends on extracellular K⁺ concentration with B_glia representing the buffering strength [15]. Two factors allow the glia to provide a nearly insatiable buffer for the extracellular space. The first is the large size of the glial network. Second, the glial endfeet surround the pericapillary space, which through interaction with arteriole walls affecting blood flow, and transport of ions to the vascular space, amplifies the effective buffering capability of the glia [71–73].

Chloride is the primary permeant anion and its homeostasis is important for a range of neurophysiological processes. Neurons regulate intracellular chloride ([Cl⁻]_i) through cation-chloride cotransporters, especially the Na⁺/K⁺/2Cl⁻ cotransporter (NKCC1) and K⁺/Cl⁻ cotransporter (KCC2) [74]. In the embryonic and early postnatal brain, neurons show robust expression of NKCC1 but minimal expression of KCC2 [74]. In the mature brain, the expression of KCC2 increases, accompanied by a concurrent down regulation of NKCC1 expression [74]. KCC2 is important in maintaining low [Cl⁻]_i, resulting in hyperpolarizing GABA responses. Since KCC2 operates close to its thermodynamic equilibrium: [Cl⁻]_i = [Cl⁻]_o[K⁺]_o/[K⁺]_i (i.e. E_Cl = E_K) [74], even a small increase in [K]_o in the mature brain will reverse Cl⁻ transport, from efflux to influx. [K]_i, [Cl]_i, and [Cl]_o are intracellular K⁺, Cl⁻, and extracellular Cl⁻ concentrations respectively.

KCC2 in our model is formulated by a logarithmic function [26, 33] (6) where ρ_KCC is the maximum strength of KCC2 and estimated using the peak conductance given in [75]. In [33], we also included the Na⁺/K⁺/2Cl⁻ (NKCC1) cotransporter, however, the results in this paper qualitatively remain the same without NKCC1 and hence it is excluded from the model. Nevertheless, NKCC1 should be included while modeling neurons from the embryonic or early postnatal brain [33, 74] and perhaps from pathological conditions where there may be aberrant transporter expression [76].

Intracellular Na⁺ concentration is controlled by transient Na⁺, Na⁺ leak, and Na⁺ − K⁺ pump currents [15] (7)

Previously, we assumed that the flow of Na⁺ into the cell is compensated by the flow of K⁺ out of the cell so that [K]_i can be approximated by [15] (8)

We also assumed that the total amount of sodium is conserved, and hence only one differential equation for sodium is needed [15], so that (9) where 140mM, 18mM, and 144mM in the above equations are the normal resting values of [K]_i, [Na]_i, and [Na]_o respectively.

The intracellular Cl⁻ concentration is modeled by Cl⁻ leak and K⁺ − Cl⁻ cotransporter [33] (10) and the extracellular Cl⁻ concentration, [Cl]_o, is set according to the conservation of charge due to Na⁺, K⁺, Cl⁻, and Ca⁺ ions in the extracellular space i.e. (11) Where [Ca]_o = 1mM is the extracellular calcium (Ca²⁺) concentration.

The main conclusions in this paper qualitatively remain the same when [K]_i, [Na]_o, and [Cl]_o are modeled by rate equations governed by various fluxes (see “Results” section) instead of the simplified version due to the conservations shown above. This is consistent with our previous findings where we demonstrated that the unification of SZ and SD is qualitatively preserved when using simplified conservation equations [33]. However, the simplification due to conservations is helpful in performing bifurcation analysis using numerical solvers based on continuation methods such as XPPAUT [77]. Such simplifications are common and especially important in neuronal models where multiple time-scales are involved as in our model [15, 33, 78].

The reversal potentials for K⁺, Na⁺ and Cl⁻ are updated based on the instantaneous ion concentrations using the Nernst equations (12)

The dynamic ion concentrations feedback into the Hodgkin-Huxley type Eqs (1, 2) through Eq (12). A more complex formulation would employ Goldman-Hodgkin-Katz voltage and current equations, which were found qualitatively similar in terms of unification in recent work [33].

Dynamic cell volume

Intense neuronal firing during SZ, SD, AD and the relevant changes in ion concentrations cause cell swelling. We use the formalism of [14] to model the change in cell volume (13) where and Vol_initial are the instantaneous and initial volumes of the cell. π_o and π_i are the sums of all ion concentrations outside and inside the cell respectively, i.e. [A]_i = 132.1mM [79] and [A]_o = 18mM are the intra- and extracellular concentrations of the impermeable anions. The value of [A]_o is based on charge balance under resting conditions using [Cl]_o = 132mM, and [Ca]_i = 100nM is the intracellular Ca²⁺ concentration. Following [79], we implement the change in the cell volume as a first-order process. (14) The time-scale of volume change, τ_v, is important for the spontaneous SZ–SD transition. For the transition to happen, the recovery of the cell from the swollen state would have to be slower than the electrical response changes during a SZ. The cell would also have to be in a SZ state for long enough to produce this effect, and that lower levels of [K]_o, with shorter duration seizures, would keep the cell out of SD. If the cell were to swell and shrink very fast so that it would recover its pre-seizure volume before going into next seizure, the cell would only exhibit spontaneous periodic seizures without going through the SZ–SD transition. To ensure the robustness of our results and that the spontaneous SZ–SD transition seen in our model occurs even with faster volume changes, we considered τ_v = 50ms, significantly smaller than the previously used phenomenological value of 250ms [33, 79]. Nevertheless, we tested a wide range of τ_v values and found that the model exhibits spontaneous SZ–SD transition for 40ms ≤ τ_v ≤ 360ms. For τ_v < 40ms, the cell recovers from swelling before going into seizure for the second time. For τ_v > 360ms, cell still exhibits spontaneous SZ–SD transition, however, it seizes more than once before going into SD state due to the slower change in volume. The number of SZs before the cell transitions into SD increases as we increase τ_v. For example, for τ_v = 400ms, 500ms, and 1000ms, the cell seizes twice, thrice, and seven times respectively before making a transition to SD. While, the spontaneous SZ–SD transition itself is a robust phenomenon, τ_v > 360ms causes the cell to seize multiple times prior to the transition. The rest of the results including all bifurcation diagrams in the paper qualitatively remain the same as we change τ_v.

In simulations where intracellular volume is treated as variable, the conductance densities for Na⁺, K⁺, and Cl⁻ currents (g_Na etc) are modified so that the total conductance for a channel type over the whole cell remains fixed even though the conductance per unit area is changing (15) Where A_ss and A_ins are the steady state (initial) and instantaneous cell surface areas respectively. and g_X are the conductance densities in case of fixed and changing volumes respectively. The effect of dynamic volume changes on membrane capacitance is negligible and not included. For example, the total capacitance of the cell during the spontaneous transitions between SZ and SD shown in Fig 2 increased from initial 2.9074pF to 3.0295pF as the cell swelled, a variation that did not change the results qualitatively.

The dynamic change in volume causes the concentration of a given ion specie to change for a fixed number of ions. Thus, the dynamic volume leads to an additional flux term in the ion concentration equations. Thus the rate equation for [K]_o (Eq 3) becomes Where Vol_o is the extracellular volume. Similarly, the rate equations for [Na]_i (Eq 7) and [Cl]_i (Eq 10) also have additional terms equal to and respectively. However, these fluxes do not change the results presented in this paper qualitatively and are not considered.

Notice that there are four different time-scales in our model: fast (variables V, n, and h; Eq (1)), intermediate ([K]_o, [Na]_i; Eqs (3, 7)), slow ([O₂], [Cl]_i; Eqs (5, 10)), and infra-slow (Vol; Eq (14)). Although we focus on the implications of dynamic intracellular volume for cell behavior in pathological states in this study, we will also analyze the evolution of other slow variables during these states and how they effect cell pathology.

Numerical methods

The coupled differential equations are solved in Fortran 90 using 4th order Runge-Kutta method. The steady states in Fig 1A, and the bifurcation diagrams in Figs 5 and 6 are generated using XPPAUT software [77]. In the bifurcation diagrams of [K]_o (and other slow variables), XPPAUT could only capture the small amplitude oscillations in [K]_o due to individual membrane potential spikes. It could not capture the high amplitude low frequency oscillations due to periodic SZ and SD events. Therefore, we simulated the periodic orbits in Figs 1A, 3, S1, and S2 Figs using Fortran 90. The codes reproducing key results in the paper are given in Supporting Information Text S1.